Landing on the Moon, a spacecraft fires its rockets and comes to a complete stop just \(12 \mathrm{m}\) above the lunar surface. It then drops freely to the surface. How long does it take to fall, and what's its impact speed? (Hint: Consult Appendix E.)

When we talk about free fall motion, we're discussing a scenario where an object is moving under the influence of gravity alone, without any other forces acting on it, like air resistance or thrust. Think of it as a pure expression of gravity's pull on an object.

In the case of the lunar lander, once the rockets stop firing, it experiences free fall motion all the way down to the Moon's surface. During this descent, the only force acting on the spacecraft is the Moon's gravitational pull. Remember though, free fall doesn't mean the object is falling down at constant speed; it's actually accelerating all the time it's falling, due to the force of gravity.

The acceleration due to gravity, often denoted by the symbol 'g', is the acceleration that Earth—or any other astronomical body—imparts on objects due to its gravitational pull. This value's commonplace here on Earth is approximately 9.81 m/s², but it differs depending on the celestial body you're on.

In our exercise, the context is the Moon, where gravity is much weaker, approximately 1.6 m/s². This significantly affects how fast objects accelerate when they're in free fall. It's crucial to use the correct value for 'g' because it determines not only how long it takes for an object to fall a certain distance but also its impact speed when it lands.

Understanding equations of motion is key to predicting how an object will move under certain forces. These equations relate an object's velocity, acceleration, the time it's been moving, and the distance it's traveled. For free-falling objects, we're often concerned with two main calculations:

• How long it will take an object to fall a certain distance (time of descent)
• How fast it will be moving when it hits the ground (final velocity)

These require the correct use of kinematic equations and account for initial conditions like the starting velocity and position.

For instance, in our lunar example, we had to solve for the time it took the lander to fall 12 meters. We used the equation \(s = 0.5at^2\), which directly relates distance traveled to the acceleration and time squared. The simplicity of this particular scenario—where initial velocity was zero—allowed us to use this equation neatly. The equations of motion are a set of four equations, but choosing the correct one depends on the knowns and unknowns in the situation you're analyzing.